Let T P_2 rarr RR^3 be a transformation given by T(f(x))=[(f(0)),(f(1)),(2f(1))] (a)Then show that T is a linear transformation. (b)Find and describe the kernel(null space) of T i.e Ker(T) and range of T. (c)Show that T is one-to-one.

Maiclubk

Maiclubk

Answered question

2021-03-02

Let T P2R3 be a transformation given by
T(f(x))=[f(0)f(1)2f(1)]
(a)Then show that T is a linear transformation.
(b)Find and describe the kernel(null space) of T i.e Ker(T) and range of T.
(c)Show that T is one-to-one.

Answer & Explanation

Elberte

Elberte

Skilled2021-03-03Added 95 answers

The map T : P2R3 be a transformation given by
T(f(x))=[f(0)f(1)2f(1)].
(a)Let f,gP2 and k is a real number. Then
T(f(x)+kg(x))=T(f(f+kg)(x))
=[(f+kg)(0)(f+kg)(1)2(f+kg)f(1)]
=[f(0)+kg(0)(f(1)+kg(1))2f(1)+2kg(1)]
=[f(0)f(1)2f(1)]+[kg(0)kg(1)2kg(1)]
=T(f(x))+kT(g(x)).
Therefore T:P2R3 is a linear transformation.
(b)
Ker(T)={fP2:T(f(x))=[000]}
={fP2:[f(0)f(1)2f(1)]=[000]}
={fP2:f(0)=0,f(1)=0,2f(1)=0}
={fP2:f(0)=0,f(1)=0}
={f(x)=ax(x1),aR}
Therefore Nullity (T)=1.
Again Range(T)

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